Multicarrier modulation techniques such as orthogonal frequency-division multiplexing (OFDM), are now widely used for high speed communications over bandpass communications channels. Examples of their use include the broadcasting of digital audio and video and as the physical layer for wireless networking standards (e.g., IEEE 802.11). An OFDM system uses several low-rate sub-carriers to transmit data, where the frequencies of the sub-carriers and the symbol period are chosen so that the sub-carriers are orthogonal over the symbol period and do not interfere with one another. In an OFDM system, the data is split into N streams, which are then used to independently modulate the closely-spaced sub-carrier frequencies or tones in parallel. Practical systems use an inverse discrete Fourier transform (implemented as an inverse Fast Fourier Transform or IFFT) to generate a sampled version of a composite time-domain signal which can be converted to a signal suitable for transmission. An advantage of OFDM over single carrier modulation is the longer symbol period for a given data rate which inherently mitigates inter-symbol interference in time dispersive channels without having to resort to elaborate equalization techniques.
The basic idea of OFDM is to transmit data encoded as symbols corresponding to the constellation points of a modulation scheme such as QAM (quadrature amplitude modulation), QPSK (quadriphase shift keying), or BPSK (binary phase shift keying) on parallel modulated sub-carriers. The modulated sub-carriers are summed together to form a transmittable composite signal which can be demodulated at a receiver. The frequencies of the sub-carriers are selected so that the sub-carriers are orthogonal (i.e., uncorrelated) over the symbol period in order to allow both spectral overlap of the sub-carriers and recoverability of the symbols. Orthogonality of the sub-carriers can be achieved if the symbol period contains an integer number of cycles of each sub-carrier. An OFDM signal with minimal spacing between the sub-carriers can be produced in the following manner. Let N be the number of sub-carriers, let C[k] for k=0 to N−1 be the complex symbols which to be transmitted simultaneously, and let NT, be the OFDM symbol duration. The samples S[n] of a complex-enveloped ODFM baseband signal S(t) taken at a sampling period of Ts over one symbol interval NTs can then be expressed as:
      S    [    n    ]    =            ∑              k        =        0                    N        -        1              ⁢                  ⁢                  C        [        k        ]            ⁢              ⅇ                  (                                    2              ⁢              π              ⁢                                                          ⁢              j              ⁢                                                          ⁢              nk                        N                    )                    The above equation says that modulation of OFDM signals can be performed by an inverse discrete Fourier transform, and a discrete Fourier transform can be used to recover the C[k] symbols. The orthogonal basis functions of the discrete Fourier transform constitute the sub-carriers, and quadrature techniques can be used to generate a real-valued time-domain signal from the complex-valued exponentials. The discrete Fourier transform (DFT) and its inverse (IDFT) are preferably implemented by the fast Fourier transform algorithm (FFT).
In a typical OFDM transmitter, a serial-to-parallel converter converts a stream of input bits into N parallel streams. Groups of bits within each parallel stream are then encoded into a block of N complex-valued symbols by encoder/modulator 12, where the complex-valued symbols correspond to the constellation points of a modulation scheme such as QAM. Because these complex-valued symbols are used to determine the amplitude and phase of a particular sub-carrier, they are referred to as frequency-domain symbols. The N frequency-domain symbols, each being a complex number representing a plurality of the input bits, are next input to an N-point IDFT and converted to a serial discrete-time signal by a parallel-to-serial converter. The resulting discrete-time signal thus constitutes N samples of a time domain waveform representing a sum of orthogonal sub-carrier waveforms with each sub-carrier waveform being modulated by a frequency-domain symbol. These N samples (or the portion of an analog waveform containing the samples) may be referred to as an OFDM symbol or block. (OFDM symbols are sometimes referred to as meta-symbols, to be distinguished from the frequency domain symbols to which the input data is directly mapped.) The time domain waveform samples are then converted into an analog waveform by a digital-to-analog converter and mixed with an appropriate carrier to transmit the time-domain waveform over a communications channel. At the receiver, the time-domain waveform can then be re-sampled and discrete Fourier transformed to recover the frequency-domain symbols, which symbols are then decoded to generate the output data stream.
In actual practice, the number of frequency-domain symbols, or data subcarriers, is usually somewhat less than the total number of sub-carriers. This is done both to prevent aliasing and to provide pilot carriers which can be used for synchronization and channel equalization. Also, a guard interval is provided for each OFDM symbol in order to provide enhanced immunity to multi-path distortion. Since an OFDM symbol is the result of an IDFT, it is a periodic function and can be cyclically extended backward or forward by adding a cyclic prefix or suffix, respectively. A DFT performed on an OFDM symbol that includes samples from the guard interval but not from an adjacent OFDM symbol will simply be phased shifted from that which would be obtained by a DFT starting at the symbol boundary. Such a phase shift can be compensated for by an equalizer in the OFDM receiver. For a channel with multi-path delay spread, the received signal is a summation of the actual transmitted signal and versions of the transmitted signal with different amplitudes and delays. If the duration of the guard interval is made just larger than the maximum excess delay of the multi-path channel, no inter-symbol interference results from the multi-path delay spread.
As with any digital communications system, synchronization of an OFDM receiver with the transmitted symbols is necessary for correct demodulation. That is, the OFDM receiver must determine a trigger point for starting the DFT that results in the correct frequency-domain symbols. Such synchronization, also referred to as symbol timing recovery, may be conventionally performed using several different methods based upon, for example, embedded pilot signals or the redundancy of the guard interval. Described herein is a method and system for symbol timing recovery that is particularly suited for use in communications channels exhibiting multi-path delay spread.